Lecture 17

Complex Arithmetic

  1. History - Finding roots of quadratic equations, e.g. x2+1=0. Gauss proved the Fundamental Theorem of Algebra
  2. Complex numbers a + bi - expressible as an ordered pair (a,b) or "real" numbers
  3. Extensions
    1. Positive Integers {1,2,3,...} extended to
    2. Natural Numbers {0,1,2,3,...} extended to
    3. Integers {...,-3,-2,-1,0,1,2,3,...} extended to
    4. Rational Numbers {p/q | p,q Integers, q not 0}
    5. Irrational Numbers extended to
    6. Complex Numbers extended to
    7. Quaternions developed by William Rowan Hamilton
  4. The complex numbers form a field and a division algebra
  5. Euler's formula
    eix = cos(x) + i sin(x)
  6. Geometry of complex numbers. Microsoft uses quaternions in its DirectX v9 package. Other game developer's have used quaternions to simplify 3D geometrical transformations.